Answer:
Eulerian, not Hamiltonian
Explanation:
The given graph has 6 vertices, of which 5 are degree 2, and one is degree 4. We are to determine if it is Eulerian or Hamiltonian.
Eulerian Graph
An Eulerian graph has every vertex of even degree, and can be decomposed into cycles.
For the given graph, degrees of the vertices are either 2 or 4. The graph can be decomposed into the cycles {a, c, e, d} and {b, e, f}.
This graph meets the requirements for an Eulerian graph.
Hamiltonian Graph
For a graph to be Hamiltonian, the sum of degrees of any pair of non-adjacent vertices must be at least the number of vertices in the graph.
For the given graph, non-adjacent vertices 'b' and 'c' have a total degree of 2+2 = 4, which is less than the 6 vertices in the graph.
This graph is not Hamiltonian.